Geometry of numbers and degree bounds for rational invariants
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917424723394560 |
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| author | Blum-Smith, Ben Crane, Sylvan Guzman, Karla Menenses, Alexis Song-Hurewitz, Maxine |
| author_facet | Blum-Smith, Ben Crane, Sylvan Guzman, Karla Menenses, Alexis Song-Hurewitz, Maxine |
| contents | We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups, we also prove a new bound on the minimum degree $d$ such that the polynomials of degree $\leq d$ span the field of rational functions as a vector space over the invariant field. This latter quantity also bounds the degree $d$ such that the polynomials of degree $\leq d$ contain a copy of the regular representation of $G$, advancing an inquiry of Kollár and Tiep. The methods involve Euclidean lattices and Minkowski's geometry of numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18876 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry of numbers and degree bounds for rational invariants Blum-Smith, Ben Crane, Sylvan Guzman, Karla Menenses, Alexis Song-Hurewitz, Maxine Commutative Algebra We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups, we also prove a new bound on the minimum degree $d$ such that the polynomials of degree $\leq d$ span the field of rational functions as a vector space over the invariant field. This latter quantity also bounds the degree $d$ such that the polynomials of degree $\leq d$ contain a copy of the regular representation of $G$, advancing an inquiry of Kollár and Tiep. The methods involve Euclidean lattices and Minkowski's geometry of numbers. |
| title | Geometry of numbers and degree bounds for rational invariants |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2604.18876 |